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OPEN-ENDEDPathway 1
Becca has created an equation to model a problem. She can solve the problem by solving the equation or graphing it.
I had $5 in my savings account and then deposited $10 each week. I now have $165. How many weeks have I been saving?
10w 1 5 5 165
• Create 3 problems that can be represented using an equation of either form:
❚p 1 ❚ 5 ❚ ❚k 2 ❚ 5 ❚
• Solve each problem by solving the equation or graphing it. If you use a graph, create it on grid paper.
• Describe or show how you solved each problem.
Equation 1: ____________________
problem:
Solution:
Solving Problems Using Equations
• You can use any letter to represent a variable.
• You can use a shortcut to show multiplication. e.g., 3n 5 3 3 n
• When you solve an equation, you determine the value of the variable that makes it true.
e.g., If 3n 5 6, n 5 2.
• You can solve an equation by thinking of a balance or a length model. e.g., 10w 1 5 5 65
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165510w
• To graph equations in the form ❚p 1 ❚ 5 ❚ and ❚k 2 ❚ 5 ❚, create a table of values for 3 ordered pairs and plot them. Then extend the points in a straight line or draw a line through them.
Solving problems Using Equations, pathway 1Leaps and Bounds
Leap SR 7-8_Topic 12.indd 213 2/25/12 4:14 PM
2. Raj has read 40 pages of his book and plans to read 25 pages more each night. Use an equation to represent the problem. Then solve the problem. Show or describe your work.
a) How many pages will he have read after 10 more nights?
equation: _____________________
b) How long will it take Raj to read 690 pages?
equation: _____________________
3. Use an equation to solve each problem. Show or describe your work.
a) A rectangle that is 30 cm long has a perimeter of 92 cm. What is the width?
equation: _____________________
b) A rectangle is 30 cm long and 10 cm wide. What is the perimeter?
equation: _____________________
4. Use an equation to solve the problem below. Show or describe your work.
Alissa does yard work. She charges $18 to mow a lawn. Before she started her business, she spent $90 to repair her lawnmower. How many lawns would she have to mow to earn a profit of $450?
Leaps and BoundsSolving Simple Equations, pathway 2
Leap SR 7-8_Topic 12.indd 216 2/25/12 4:14 PM
• Write and model 4 equations that use the variable k.– One or more equations should involve subtraction.– Three or more equations should involve multiplying k
Leaps and Bounds Solving Simple Equations, pathway 2
Leap SR 7-8_Topic 12.indd 217 2/25/12 4:14 PM
GuiDEDPathway 2
You will need• pan balance,
paper bags, and base ten cubes or linking cubes
Jorge is modelling 2 equations.An equation is a statement about 2 amounts being equal. The amounts can be numbers, for example, 2 1 4 5 5 1 1,or variables representing numbers, for example, 2m 1 3 5 11.
Jorge has used a length model and a balance model to represent the 2 equations.
An equation that has a variable can be solved in different ways.
using a Length Model
• You can use a length model to solve an equation.
You can think of the 2 sides of the equation as 2 matching rectangles, one above the other.
LEAP 7/8 SR
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2nd pass
m m 3
11
2m 3 11
To solve the equation 2m 1 3 5 11, you take away a length of 3 from each rectangle and the new rectangles still match in length.
The sections labelled m are equal, so 2m 5 8, and m 5 4.
LEAP 7/8 SR
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m
11 3 8
m
2m 8, so m 4.
2m 3 3 11 3
m m 3
11
Solving Simple Equations
• You can use any letter to represent a variable.
• You can use a shortcut to show multiplication involving a variable. e.g., 3n 5 3 3 n
• When you solve an equation, you determine the value of the variable that makes it true.
Leaps and BoundsSolving Simple Equations, pathway 2
Leap SR 7-8_Topic 12.indd 218 2/25/12 4:14 PM
using a Balance Model
• You can use a balance model to solve an equation.
You can think of the equation as 2 sides with the same mass.
LEAP 7/8 TR
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2nd pass
m m
2m 3 11
To solve the equation 2m 1 3 5 11, you can take 3 cubes from each side without changing the balance. That means each bag labelled m has 4 cubes in it, so m 5 4.
LEAP 7/8 SR
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2nd pass
m m m m
2m 8, so m 42m 3 3 11 3
Guessing and Testing
• You can also solve an equation by guessing and testing.
For example, to solve the equation 2 j 2 8 5 10, you can try different values for j in the equation that make sense, until you find a value that works.
Guess Test
first guess: j 5 6 2 3 6 2 8 5 4 too low
second guess: j 5 10 2 3 10 2 8 5 12 too high
third guess: j 5 9 2 3 9 2 8 5 10 j 5 9
Try These 1. What equation does each balance model represent?
a)
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x
_______________________
b)
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x x
_______________________
• If you do not have the materials to create a balance model, a simple sketch of the model will work.
Leaps and Bounds Solving Simple Equations, pathway 2
Leap SR 7-8_Topic 12.indd 219 2/25/12 4:14 PM
2. What equation does each length model represent?
a)
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10
1x xx
_______________________
b)
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1st pass
10
2xxxx
_______________________
3. Sketch a balance or length model to represent each equation.Do not solve it.
a) j 1 2 5 10 c) 2 j 1 2 5 10
b) k 1 3 5 11 d) 3 j 1 4 5 13
4. Use a model to help explain why each statement is true.
a) If 3x 1 5 5 11, then 3x 5 11 2 5. b) If 2x 5 14, then x 5 7.
5. You can add 1 to each side of the equation 2x 2 1 5 11 without changing the value of x. Use this length model to show or explain why this is possible.
Leaps and Bounds Solving Simple Equations, pathway 2
Leap SR 7-8_Topic 12.indd 221 2/25/12 4:14 PM
OPEN-ENDEDPathway 3
Renée is modelling the algebraic equation 2 1 3v 5 11 with algebra tiles. The equation means, “What number, when tripled and added to 2, results in 11?”
In the equation 2 1 3v 5 11, the variable v represents a single unknown amount. When you solve the equation, v 5 3.
Variables are also used to create equations that describe relationships among quantities. For example, the equation j 5 3 1 2w means,“The value of the variable j is 3 more than double the value of the variable w.”
In an equation that represents a relationship, there are multiple possible values for the variables that make it true. For example, in j 5 3 1 2w, if w 5 1, then j 5 5. If w 5 2, then j 5 6.
The equals sign in an equation means the algebraic expressions on both sides of the equation have the same value.
In 2 1 3v 5 11, 2 1 3v has the same value as 11.
In j 5 2 1 2w, j has the same value as 2 1 2w.
• Write an equation that uses a variable to represent a single unknown amount. Describe or show what your equation means. Solve the equation to determine what amount the variable represents.
equation: _______________________
Using Variables
algebraic equationa statement that uses an equals sign to show that 2 quantities have the same value
algebraic expressiona phrase that involves a set of numbers, operation signs, and variables
• Any letter can be used to represent a variable without changing the meaning of the expression or equation.
• You can use a shortcut to show multiplication involving a variable. e.g., 3n 5 3 3 n.
• When you solve an equation, you determine the value of the variable that makes the equation true. e.g.,
• Change one thing in your equation to make a new equation. Describe what your new equation means. Solve the equation.
new equation:
• Change something else in your equation to make a new equation. Describe what your new equation means. Solve the equation.
new equation:
• Write an equation that uses variables and represents a relationship. Describe what your equation means. Solve the equation to determine 2 sets of possible values for the variables.
equation: _______________________
• Change one thing in your equation. Describe what your new equation means. Solve the equation to determine 2 sets of possible values for the variables.
new equation:
• Change something else in your equation. Describe what your new equation means. Solve the equation to determine 2 sets of possible values for the variables.
new equation:
• Write an algebraic expression that is always greater than 3n 1 2 if n is a whole number. Substitute 3 values into the expression 3n 1 2 and into your expression to see if it seems right.
expression: _______________________
substituteto replace the value of a variable with a specific amount
Mathematicians use letters as variables to represent unknown values. Steig is modelling an algebraic equation that uses a variable.
• You can use a variable to represent a single unknown value.
Recall equations such as 4 3 ❚ 5 12, which means,“If you multiply a number by 4, you get 12.”
You can use a letter variable instead of ❚ to represent the unknown number: 4n 5 12 means the same thing as 4 3 ❚ 5 12.
When you solve the equation 4n 5 12, n 5 3.
• You can also use variables to create an equation that represents a relationship. In a relationship, there is more than one set of values that make the equation true.
Suppose a rectangle is 10 cm long, but you don’t know its perimeter or width. You could represent the relationship using the equation P 5 10 1 w 1 10 1 w.
You could also use P 5 20 1 2w, because the expression 10 1 w 1 10 1 w is equivalent to 20 1 2w.
There are multiple possible values for the variables that make a relationship true.
For example, when you solve P 5 20 1 2w for w 5 2, P 5 24. If w 5 5, then P 5 30.
• Each equation above has an equals sign that means the algebraic expressions on both sides of the equation have the same value.
In 4 1 n 5 12, 4 1 n has the same value as 12.
In P 5 20 1 2w, P has the same value as 20 1 2w.
Using Variables Pathway 3
You will need• algebra tiles: x-tiles
and 1-tiles
• You can use any letter to represent a variable without changing the meaning of the expression or equation.
• You can use a shortcut to show multiplication involving a variable. e.g., 3n 5 3 3 n
• When you solve an equation, you determine the value of the variable that makes the equation true. e.g.,
If 3n 5 6, then n 5 2. If j 5 2w and w 5 1,
then j 5 2. If w 5 2,then j 5 4, and so on.
Remember
algebraic equationa statement that uses an equals sign to show that 2 quantities have the same value
algebraic expressiona phrase that involves a set of numbers, operation signs, and variables
• You can model algebraic expressions using algebra tiles.
Each rectangle tile represents the variable, and each square tile represents 1. Light grey tiles show positive amounts, and dark grey tiles show negative amounts. The tile model below shows 2w 1 20.
LEAP 7/8 SR
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Pass
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D. Loates
2nd pass
2w 20
The model shows why you cannot just combine 2w and 20 to determine the sum of 2w 1 20, because 2w and 20 are represented by different-sized tiles.
• You can use algebraic expressions to solve problems.
To calculate the perimeter of a 10 cm long rectangle, you can substitute a value for the width (w ) into the expression 20 1 2w.
For a width of 2 cm (w 5 2), the perimeter is 10 1 2 1 10 1 2 5 24 cm.
For w 5 5, it is 10 1 5 1 10 1 5 5 30.
• How do you know that the equation w 1 3 5 w 1 2 1 1 represents a relationship?
Try These 1. Are the variables used to represent an unknown value in the equation
or are they used in an equation that to describes a relationship? Circle the correct answer.
a) 30 4 k 5 6 unknown value relationship
b) 4 3 3 5 m unknown value relationship
c) j 1 10 5 5 1 j 1 5 unknown value relationship
d) m 5 10 2 n unknown value relationship
e) 5 5 10 2 r unknown value relationship
2. Match each expression to a meaning. One expression has no match.
half of a number
the difference between 20 and half of a number
20 more than double a number
2 more than 20 times a number
the difference between 20 and double a number
2n 1 20
20 2 2n
20 4 2 2 2n
n 4 2
20 2 n 4 2
20n 1 2
substituteto replace the value of a variable with a specific amount